Heavy-organic particle deposition from petroleum fluid flow in oil wells and pipelines

Transcription

1 etroleum cience olume 7, ages 5-58, 5 OI.7/s Heavy-organic particle eposition from petroleum flui flow in oil wells an pipelines Joel scobeo an G. Ali Mansoori University of Illinois at Chicago, 85. Morgan t. (M/C 63, Chicago, IL UA China University of etroleum (eijing an pringer-erlag erlin Heielberg Abstract: uspene asphaltenic heavy organic particles in petroleum fluis may stick to the inner walls of oil wells an pipelines. This is the major reason for fouling an arterial blockage in the petroleum inustry. This report is evote the stuy of the mechanism of migration of suspene heavy organic particles towars the walls in oil-proucing wells an pipelines. In this report we present a etaile analytical moel for the heavy organics suspene particle eposition coefficient corresponing to petroleum fluis flow prouction conitions in oil wells. We preict the rate of particle eposition uring various turbulent flow regimes. The turbulent bounary layer theory an the concepts of mass transfer are utilize to moel an calculate the particle eposition rates on the walls of flowing conuits. The evelope moel accounts for the ey iffusivity, an rownian iffusivity as well as for inertial effects. The analysis presente in this paper shows that rates of particle eposition (uring petroleum flui prouction on the walls of the flowing channel ue solely to iffusion effects are small. It is also shown that eposition rates ecrease with increasing particle size. However, when the process is momentum controlle (large particle sizes higher eposition rates are expecte. Key wors: Asphaltene, rownian eposition coefficient, iffusivity, iamonois, heavy organic particles, paraffin/wax, particle eposition, petroleum flui, prefouling behavior, prouction operation, suspene particles, turbulent flow Introuction A common problem face by the oil inustry is the eposition of heavy organics insie prouction wells, storage vessels, an transfer pipelines. Our stuies an experiences have inicate that heavy organic eposition is one of the major factors that increase the cost of prouction an transportation of petroleum fluis (Mansoori, 988; Carpentier et al, 7. Furthermore, miscible flooing of petroleum reservoirs by lean gas, carbon ioxie, natural gas, an other high pressure injection fluis has become an economically viable technique for petroleum prouction. Introuction of a miscible flui in petroleum reservoirs, will, in general, prouce a number of alterations in petroleum flui flow an phase behavior an reservoir rock characteristics. One such alteration is the heavy organic precipitation, flocculation an eposition (asphaltenes, iamonois, etc., which in most of the observe instances result in plugging or wettability reversal in the conuits (scobeo an Mansoori, 995a; 995b; ranco et al, ; Mousavi-ehghani et al, 4; Mansoori et al, 7. resent aress: Case Western Reserve University, 9 ucli Ave, Clevelan, OH joelescobeo@gmail.com Corresponing author. mansoori@uic.eu In our recent reports we presente the various causes an effects of phase behavior of petroleum fluis containing heavy organic fractions an their epositions (Mansoori, 9a; 9b. It is always preferre to prevent heavy organics eposition uring petroleum flui flows. In cases when heavy organics precipitation can not be prevente we nee to unerstan the flui flow behavior which is less likely to cause fouling of a conuit. In these cases there is a nee for unerstaning how the precipitate an flocculate particles suspene in the oil will behave uner certain flow conitions. This motivate the research presente in this report. Our main objective is to stuy the behavior of suspene heavy organic particles uring flow conitions. As a preliminary step, the tenency of the crue oil to form soli particles, whenever a miscible solvent is injecte into the petroleum reservoir, must be etermine. This may be accomplishe by using existing experimental techniques (Mousavi-ehghani et al, 4; Mansoori et al, 7 together with preictive moels an packages (ranco et al,. These combine experimental/preictive approaches have proven to be a useful tool for esign of prouction an transportation schemes for crue oils prone to heavy organics precipitation, flocculation an eposition. The stuy of the behavior of suspene heavy-organic particles uring flow conitions has been focuse on the prouction well since flow through a pipeline is only a special

2 J. scobeo an G.A. Mansoori Heavy-organic particle eposition from petroleum flui flow in oil wells an pipelines etroleum cience 7, 5-58, 53 case of this more general one. A typical prouction well may be ivie into two istinct sections (ee Fig. : (I The pressure region above the bubble point (singlephase is the emphasis of the present report. Unerstaning of eposition of particles in this region is especially important for particles which enter into the oil well from the reservoir. (II The analysis of the region below the bubble-point pressure (two-phase flow is the subject of our ongoing research. Annularmist flow lug flow Transition flow ubble flow ubble point ingle-phase turbulent flow Oil reservoir ublaminar layer Turbulent core Transition (buffer region Fig. Illustration of velocity istribution an ifferent flow regimes in the prefouling single-phase turbulent flow conition in the oil well In this report we present an analytical moel for the heavy organic suspene particle eposition coefficient prior to eposition corresponing to petroleum fluis flow in proucing wells conition. In our previous publications (scobeo an Mansoori, 995a; 995b; we reporte various segments of this analytical moel some of which containe typographical errors. For the sake of comprehensiveness an to provie the reaership with the etails of the algebraic manipulations involve we report here the etails of the moel in its entirety. evelopment of the analytical moel The theory presente here is for a turbulent petroleum flui system of constant ensity an viscosity. For petroleum flui flow in wells this is the case for the region above the bubble pressure where only the liqui phase is the meium for the suspene particles. ubstantial work has been one by many researchers on the topic of particle eposition on the walls of channels or pipes in turbulent flow by many researchers (Lin et al, 953; Laufer, 954; Frielaner an Johnstone, 957; eal, 97; Chen an Ahmai, 997; erevich an Zaichik, 988; Johansen, 99. The moel presente here is a combination of the work performe by the authors mentione above, moifie to be applicable to the eposition of heavy organic particles in turbulent petroleum flui flow. A key assumption in the evelopment of this moel is that fully evelope petroleum turbulent flow has a structure as propose by Lin et al. From experimental observations, they propose a generalize velocity istribution for turbulent flow of fluis in pipes comprise of three main regions: - A sublaminar (wall layer r 5, - A buffer layer 5 r 3, - A turbulent core 3 r, where r = ( f / / ν is a imensionless istance measure from the wall. This imensionless istance is a function of the inner pipe iameter, i, flui average velocity,, the Fanning friction factor, f, an the flui kinematic viscosity, ν. The moel presente here is for a system of constant ensity an viscosity. Therefore, it is applicable only to a single phase petroleum flui flow. This theory can be extene to the region below the bubble point pressure (gasliqui slug flow, etc. using reliable expressions for petroleum viscosity an ensity versus pressure an temperature. The assumption of constant viscosity an constant ensity is justifie since ensity changes are not appreciable until the bubble point pressure is reache insie the well (or tubing. It is also assume that suspene particles are of uniform iameter, an that, ue to their rather low concentration we can neglect particle-particle interactions. We assume the thickness of the bounary layer is quite small compare to the raius of the pipe as a result we can neglect the wall curvature effects in our moel. We start with reporting the equation which is use to escribe the particle flux, N, in terms of the iffusivities an the concentration graient (on Karman, 4, i.e., ( N = + C r where is the rownian iffusivity; is the ey iffusivity; C is the particle concentration; an r is the raial istance. The rownian iffusivity is efine by the following equation: kt = 3π μ where k is the oltzmann constant (k = J/K; T is the absolute temperature; is the particle iameter; an µ is the liqui viscosity of the suspening meium (petroleum flui. quation ( is subject to the bounary conition: at r = C C = where C is the particle concentration at r = an is the particle stopping istance measure from the wall. A particle nees to iffuse only within one stopping istance ( (

3 54 J. scobeo an G.A. Mansoori Heavy-organic particle eposition from petroleum flui flow in oil wells an pipelines etroleum cience 7, 5-58, from the wall, an from this point on, ue to the particle momentum, it woul coast to the wall. For small particles the stopping istance is small compare with the bounary layer thickness an consequently iffusion ominates. The propose correlation for the particle stopping istance is (Frielaner an Johnstone, 957:.5 ρ f / (3 = + μ where ρ is the ensity of particles; is the average velocity of petroleum fluis; an f is the Fanning friction factor. quation ( may be integrate following the proceure for the calculation of temperature rop across a composite wall. We will fin the concentration profiles from point to point across the bounary layer. That is, we will calculate the concentration ifferences through the sublaminar layer, the buffer region an the turbulent core. y aing these concentration ifferences we can fin the overall particle flux in terms of the average an wall concentrations. efore we integrate quation ( for C, we nee to have expressions for N an as functions of the raial istance (r for each of the three main regions in the oil well an pipeline, i.e. sublaminar layer, buffer layer an turbulent core. ublaminar Layer Johansen (99 propose the following correlation to express the ey iffusivity as a function of raial istance (r for the sublaminar layer: r r = v for r 5 or =.5 ν.5 (4 In this equation v is the kinematic viscosity of the flowing petroleum flui, r is the imensionless raial istance an 5 represents the sublaminar layer (r 5 ey iffusivity. The particle molar flux, N, is assume to vary linearly from the wall to the center line of the channel, as propose by eal (97: r N = N (5 o In this equation N o is the particle flux at the wall; r is a imensionless raial istance an i is the imensionless inner well (or tubing iameter, both efine by the following equations: r = r ν f / (6 following equation: f / (8 = ν For the sublaminar layer quation ( may be integrate following the proceure for the calculation of temperature rop across a composite wall. We will fin the concentration profiles from point to point across the bounary layer. That is, we will calculate the concentration ifferences through the sublaminar layer, the buffer region an the turbulent core. y aing these concentration ifferences we can fin the overall particle flux in terms of the average an wall concentrations. Note that quation (4 is only vali for imensionless raial istances smaller than 5, which is the limit of the sublaminar layer. Introucing all the new imensionless variables an the expressions for N an into quation (, consiering that r = r v we get: f / N N f 3 r r C = o = + / ν.5 r subject to the following bounary conitions: at r = C = C at 5 r = C = C5 Rearranging quation (9, integrating an applying the above bounary conitions we arrive at the following integral form: /3 /3 N.5N (.5 N ch ch C5 C = F F 3 3 f / ( In the above equation N ch is the chmit number efine as: ν N ch an F an F are efine by the following expressions: + F = ln + ( 5 ( + 5φ ( φ ln φ ( φ φ φ 3 tan 3 tan 3 3 (9 ( ( f / = ν where i is the inner iameter of the well (or tubing; is the average flui velocity; f is the Fanning friction factor; an v is kinematic viscosity of the flowing flui, m /s. Let us also efine the imensionless stopping-istance, by the (7 ( φ ( φ ( 5 φ F = ln ln + ( + 5φ + φ φ 3 tan 3 tan 3 3 where for simplicity we have efine: (3

4 J. scobeo an G.A. Mansoori Heavy-organic particle eposition from petroleum flui flow in oil wells an pipelines etroleum cience 7, 5-58, 55 /3 Nch φ.5 (4 quations (-4 escribe the transport of suspene particles in the sublaminar layer to the wall in terms of the concentration ifference between the limits r = s (imensionless stopping istance an r =5 (limit of the sublaminar layer. uffer Layer The next step is the calculation of the particle flux between the concentration at r =5 an r =3 (limit of the buffer layer. The ey iffusivity expression for the buffer layer is assume to be: r ν = r ν =.93.4 for 5 r 3 or (5 Integration of quation (, using quation (5 for gives: 3 C3 C5 No = [.4θ F3 ] ln f /.4 5 with.4 3θ.4 5θ F3 = ln ln.4 + 3θ.4 + 5θ θ N ch (.93 Nch /. (6 (7 (8 quation (6 escribes the particle transport in terms of the concentration ifference between the limits of the buffer layer. 3 Turbulent core The following step is the calculation of the particle transport rate in the turbulent core in terms of the ifference between the concentration at r =3 (upper limit of the buffer layer an the bulk concentration (average conc.. The ey iffusivity for the turbulent core is taken to be (Johansen, 99: 3 =.4r ν for r 3 or =.4r (9 ν If we assume that at = we have C = C, then we can integrate quation ( using quation (9 to obtain the following expression for C, C o C3 N = f /.5 +.4r N ch 5r 5r.5 + ln + N ch.4 rn + ch ( In this expression r is the imensionless raial istance (measure from the wall where =. quation ( escribes the particle transport in terms of the concentration ifference between the bulk (average an the upper limit of the buffer layer. o far, we have expressions for the three ifferent regions (sublaminar/wall layer, buffer layer, an turbulent core. Now they may be ae together to obtain an expression for N in terms of C, C, average flui velocity,, an physical parameters of the system. Until now, only imensionless stopping istances ( with values less than 5 have been consiere. However, for particles large enough coul be greater than 5. If so, then the preceing analysis is not vali uner these conitions. This ifficulty may be overcome if quation ( is integrate between the limits C C an = at C C 3 = at r = r =3, using the ey iffusivity correlation for the buffer layer as expresse by equation (5. Introucing quation (5 into quation ( an integrating using the assumptions note previously, we get: C C F f /.4 uch that,.4n = s θ 4 ln i.4 3. θ.4 θ F4 = ln ln.4 3. θ θ + ( ( an θ is efine by q. (8. For =5 then quations ( an ( reuce to quations (6 an (7. quation (3 still applies to the turbulent core. For particles with a imensionless stopping istance <5 We a quations (, (6, an (, to obtain an expression for the mass transfer (transport coefficient efine as N /( C C = K with the imension of velocity in cm/sec. The expression for the eposition coefficient obtaine is: φ.5 φ K = f / F F +.4θ F3 + F5 3.5 (3 arameter F, F an F 3 appearing in this equation are the same as efine previously by quations (, (3, an (7, respectively an parameter F 5 is given below:

5 56 J. scobeo an G.A. Mansoori Heavy-organic particle eposition from petroleum flui flow in oil wells an pipelines etroleum cience 7, 5-58, F = ln (.5 r φ 5 5r.5 + ln +.5 φ + (.5 φ (4 For particles with a imensionless stopping istance 5 <3: y combining quations ( an ( an solving for K we get: { θ } K = f / /.4 F + F 4 5 (5 arameters F 4 an F 5 appearing in this equation are the same as previously efine by quations ( an (4, respectively. Inertial effects In the above analysis we have erive analytical expressions for the mass eposition coefficient for ifferent particle sizes in terms of the imensionless stopping istance. Next we must account for the inertial effects. We use the following expression to account for inertial effects (eal, 97: K N K p ϑ = = C K + p ϑ iner avs (6 iner where K is the eposition coefficient (mass transfer coefficient which contains the inertial effects; N is the particle flux at the wall (as previously efine; p is the particle sticking factor (taken equal to unity; ϑ is the average velocity of the particles near the walls, an it consists of two parts: ϑ = ϑ + ϑ (7 where ϑ is the particle velocity component ue to flui motion an ϑ is the component ue to particle rownian motion, / kt = ϑ (8 3πm with k = f/ ( ϑ ( / + ϑ ( 4 where m is the particle mass; ϑ ( / is the particle velocity at a imensionless raial istance /; is the imensionless particle iameter; an ϑ ( is the particle velocity at a imensionless raial istance (imensionless stopping istance. The quantity ϑ can be calculate using the following correlation propose by Laufer (954: ϑ =.5r for r ϑ =.5r +.5(r for r 3 The particles can be anywhere between r = / an r =. Consiering this we are to choose the appropriate expression for ϑ. The analysis for particle eposition onto the walls of a flowing channel from turbulent flui streams is conclue by taking into account the inertial effects as in quation (5. At this point all the phenomena influencing the eposition rate (rownian iffusivity, ey iffusivity, an inertial effects have been taken into account. 3 reictions for particle eposition in wells an tubings In our previous publications (scobeo an Mansoori, 995a; 995b; we compare the preictions of the propose moel with the available experimental ata an the other moels. ince experimental ata for particle eposition from turbulent petroleum flui flows were scarce we use the eposition ata from aerosols to verify our moel. As it was shown, the results of our analysis for particle eposition from turbulent flui streams were in very goo agreement with the experimental eposition rates for aluminum an iron particles in air. The preictions of the present moel showe a better agreement with the mentione experimental ata than the moels propose earlier. We believe our moel is suitable for a priori preictions of the particle eposition coefficient from turbulent petroleum flui flow. Fig. shows the preicte eposition coefficients for particles as a function of particle iameter suspene in a light petroleum flui with 3. AI gravity (corresponing to.875 G an the kinematic viscosity of centi-tokes (ct at various prouction rates. Fig. 3 shows the preicte eposition coefficients values for petroleum fluis with ifferent kinematic viscosities (an at ifferent AIs an containing suspene particles of, nm in iameter. eposition coefficient, cm/min I- I-4 I-6 I-8 article iameter, nm Fig. ffect of particle size on the eposition coefficient for a 3. AI petroleum flui with a kinematic viscosity of ct at various prouction rates in m 3 /ay The particle sizes analyze range from 5 to, nm. Accoring to Fig., eposition coefficients are generally small except at high prouction rates an for very large particles. Note a minimum after which the eposition coefficient increases more rapily with increasing particle iameter. This is ue to the fact that at the minimum point the eposition process is momentum controlle. Accoring to Fig. 3, particle eposition from turbulent petroleum flui flow is very small when the iameter of the suspene particles

6 J. scobeo an G.A. Mansoori Heavy-organic particle eposition from petroleum flui flow in oil wells an pipelines etroleum cience 7, 5-58, 57 eposition coefficient, cm/min I- I-4 I-6 Kinematic viscosity.9ct 3.7ct 4.97ct.6ct 3.ct I-8 3 Oil rouction rate, m 3 /ay Fig. 3 ffect of petroleum prouction rate on the eposition coefficient of micron ( nm iameter suspene particles in petroleum fluis of various kinematic viscosities ranging from.9-3. ct is less than, nm. Fig. inicates a ecrease in the eposition coefficients with increasing kinematic viscosity an an increase in the eposition coefficient with increasing prouction rate. The lighter the petroleum flui, the higher the probability of having particle eposition. Fig. 4 shows the preicte eposition coefficients for particles as a function of the crue oil prouction rate for two extreme particle sizes, 3.5 nm (corresponing to asphaltene steric-colloial particles in original crue oil an 7 nm (corresponing to flocculate asphaltene aggregates. As it was mentione the present analysis is for the single-phase region of the well-tubing (region before the bubble point pressure is reache. One can notice from this figure the very small values of eposition coefficients for 3.5 nm particles, for the volumetric flow rates of up to 3, m 3 /ay. Fig. 4 also inicates that eposition coefficients increase with increasing velocities. However, when the oil prouction rate is about, m 3 /ay, eposition coefficient is roughly.6 cm/min. Therefore we woul expect sizeable amounts of eposition at high prouction rates. eposition coefficient K 3, cm/min.. Oil prouction rate, m 3 /ay article size 3.5 nm 7 nm Fig. 4 reicte eposition coefficients as a function of the crue oil prouction rates. It is calculate for particles with a iameter of 3.5 nm suspene in a 37.8 AI petroleum flui With the iea in min of fining another factor that coul increase the tenency of the particles to eposit, we examine the effect of particle size on eposition coefficient. This is shown in Fig. 5 where we can notice that eposition coefficients ecrease with increasing particle size ue to the fact that rownian iffusion is inversely proportional to the particle size of the iffusing species. However, with increasing particle size (or mass the momentum increases, so that particle momentum effects become more important. It is expecte a minimum eposition rate for a given particle size, beyon which the process is essentially momentum controlle, resulting in higher eposition rates. For a raius of 7 nm, we still notice very small values (see Fig. 4, this means that heavy organic particle size must become much larger in orer to exhibit high eposition rates. eposition coefficient K 3, cm/min Oil prouction rate, m 3 /ay article size Fig. 5 ffect of particle size on eposition coefficient for a 37.8 AI petroleum flui nm 3nm 4nm 5nm 6nm We performe moel preictions varying the kinematic viscosity of the crue oil to stuy the effect of this parameter on the eposition coefficient. This was one because kinematic viscosity ecreases with increasing temperature, an because it is a function of the AI gravity of the crue oil. Fig. 6 shows the preicte values, an we can see a ecrease in the eposition coefficient with increasing kinematic viscosity. This means that the lighter the petroleum flui is the higher the probability of having heavy organic eposition will be. However, these preicte values are still very small. eposition coefficient K 3, cm/min 3 Kinematic viscosity Oil prouction rate, m 3 /ay.3ct.4ct.5ct.6ct.7ct Fig. 6 ffect of kinematic viscosity on eposition coefficient

7 58 J. scobeo an G.A. Mansoori Heavy-organic particle eposition from petroleum flui flow in oil wells an pipelines etroleum cience 7, 5-58, 4 Conclusions The moel evelope for the particle eposition onto the walls of a pipe from a turbulent petroleum flui stream in oil wells an pipelines was use to preict the eposition coefficient from turbulent flow prouction operations. The effect of particle size on the eposition coefficient was investigate, fining that when the eposition process is iffusion-controlle (particles with a iameter <, nm the preicte values are very small. However, when the eposition process is momentum controlle (particles with a iameter >, nm the preicte values for the eposition coefficient increase more rapily with increasing particle iameter. We also investigate the effect of petroleum flui kinematic viscosity on the eposition coefficient. We foun that the eposition coefficient ecreases with increasing petroleum flui kinematic viscosity. For kinematic viscosity of.6 ct the preicte eposition coefficient is negligible for suspene particles of, nm. We also foun that the eposition coefficient increases with increasing prouction rate. This is ue to the fact that at larger prouction rates the amount of ey iffusion is bigger. The propose moel can be use for various cases of the behavior of heavy organics particle eposition from turbulent petroleum flows so long as the particles are neutral, their sizes are stable, there are no particle-particle interactions an there are no phase transitions occurring in the flow. However, in cases where such changes are occurring in the system this moel will require appropriate moifications as presente an applie elsewhere (Mansoori G A, AHRAC: A comprehensive package of computer programs an atabase which calculates various properties of petroleum fluis containing heavy organics. eu/~mansoori/ahrac_html. Acknowlegements The authors woul like to thank Aly Hamoua, ogo Melo aes, aulo Ribeiro, Kamy epehrnouri an Mahy hirel for taking time to rea the manuscript an suggesting very useful corrections. To receive the executable computer package an the set of relate equation for our propose moel please contact the corresponing author. References ea l K. eposition of particles in turbulent flow on channel or pipe walls. Nuclear cience an ngineering : - ra nco A M, Mansoori G A, e Almeia Xavier L C, et al. Asphaltene flocculation an collapse from petroleum fluis. Journal of etroleum cience an ngineering.. 3: 7-3 Car pentier, Wilhelms A an Mansoori G A. Reservoir organic geochemistry: rocesses an applications. Journal of etroleum cience an ngineering (3-4: Che n K an Ahmai G. eposition of particles in a turbulent pipe flow. Journal of Aerosol cience (5: er evich I an Zaichik L I. article eposition from a turbulent flow. Flui ynamics (5: 7-79 sc obeo J an Mansoori G A. oli particle eposition uring turbulent flow prouction operations. aper 9488 presente at rouction Operations ymposium, -4 April 995a, Oklahoma City, Oklahoma sc obeo J an Mansoori G A. Asphaltene an other heavy-organic particle eposition uring transfer an prouction operations. aper 367 presente at Annual Technical Conference an xhibition, -5 October 995b, allas, Texas sc obeo J an Mansoori G A. refouling behavior of suspene particles in petroleum flui flow. scientia Ir., Transactions C: Chemistry an Chemical ngineering.. 7: (to appear Fri elaner K an Johnstone H F. eposition of suspene particles from turbulent gas streams. Inustrial an ngineering Chemistry (7: 5-56 Joh ansen T. The eposition of particles on vertical walls. International Journal of Multiphase flow (3: Lau fer J. The structure of turbulence in fully evelope pipe flow. NACA 74, National Avisory Committee for Aeronautics, 954 (Available from NAA as TR-74. Lin C, Moulton R W an utnam G L. Mass transfer between soli wall an flui streams. Inustrial an ngineering Chemistry (3: Man soori G A. Asphaltene eposition: An economic challenge in heavy petroleum crue utilization an processing. OC Review Man soori G A, azquez an hariaty-niassar M. olyispersity of heavy organics in crue oils an their role in oil well fouling. Journal of etroleum cience an ngineering (3-4: Man soori G A. A unifie perspective on the phase behaviour of petroleum fluis. International Journal of Oil, Gas an Coal Technology. 9a. (: 4-67 Man soori G A. hase behavior in petroleum fluis, petroleum engineering ownstream section of encyclopeia of life support systems. 33 pages. UNCO, UN, aris, France, 9b Man soori G A. AHRAC: A comprehensive package of computer programs an atabase which calculates various properties of petroleum fluis containing heavy organics. eu/~mansoori/ahrac_html Mou savi-ehghani A, Riazi M R, afaie-efti M an Mansoori G A. An analysis of methos for etermination of onsets of asphaltene phase separations. Journal of etroleum cience an ngineering (-4: on Karman T. Aeroynamics: electe Topics in the Light of Their Historical evelopment. 4 (ite by un Yanhua